AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be positive real numbers such that
. The probability that Club Truncator will finish the season with more wins than losses is 
, where 
and 
are relatively prime positive integers. Find 
.
marks. All bar of the 
possible ordered triples have been seen in that period.
be an acute triangle, and let 
be a variable interior point on the minor arc 
of its circumcircle. Let 
and 
be the feet of the perpendiculars from to lines 
and 
, respectively. Let 
be the intersection of line 
and the perpendicular from 
to 
. Let 
be the line through parallel to 
. Prove that as varies along minor arc , the line always passes through a fixed point. (Specifically: prove that there is a point 
, determined by triangle , such that no matter where is on arc , line passes through .) such that there exists a polynomial of degree with real coefficients 
whose graph contains 2025 distinct points that are the vertices of a 2025-sided regular polygon. are the points on the cartesian plane of the form 
.
be a right triangle with 
, 
, and 
. Let 
be the inscribed circle. Construct 
with 
on 
and 
on 
, such that is perpendicular to and tangent to . Construct 
with 
on 
and 
on such that is perpendicular to and tangent to . Let 
be the inscribed circle of 
and 
the inscribed circle of 
. The distance between the centers of and can be written as 
. What is ?
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
such that for all 
, 
hold:
such that for all 
be an isosceles triangle with 
and circumcircle 
. The point 
lies on the shorter arc of over the chord and is different from and 
. Let 
denote the intersection of 
and 
. Prove that the line through and is a tangent of the circumcircle of the triangle 
. with even number of divisors which can be paired such that the sum of numbers in every pair is a power of three.
if it is not parallel to any of the –axis, the 
–axis, or the line 
.
be a given integer. Determine all nonnegative integers such that there exist distinct lines in the plane satisfying both of the following:
and 
with 
, the point 
lies on at least one of the lines; and of the lines are sunny.
is a positive divisor of other than itself.
consists of positive integers, each of which has at least three proper divisors. For each 
, the integer 
is the sum of the three largest proper divisors of 
.
.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
such that:
For all real numbers 
.
